Asymptotic eigenvalue distributions of block-transposed Wishart matrices

TL;DR

This paper investigates the asymptotic eigenvalue distribution of block-transposed Wishart matrices, revealing that as dimension grows, their scaled law converges to a free difference of free Poisson laws, with implications for quantum information theory.

Contribution

It provides the first analysis of eigenvalue distributions of block-transposed Wishart matrices and characterizes conditions for their positivity support.

Findings

Eigenvalue distribution converges to a free difference of free Poisson laws as dimension increases.

Derived necessary and sufficient conditions for the measures to be supported on the positive half line.

Results have implications for quantum information theory, especially in entanglement detection.

Abstract

We study the partial transposition ${W}^\Gamma=(\mathrm{id}\otimes \mathrm{t})W\in M_{dn}(\mathbb C)$ of a Wishart matrix $W\in M_{dn}(\mathbb C)$ of parameters $(dn,dm)$. Our main result is that, with $d\to\infty$, the law of $m{W}^\Gamma$ is a free difference of free Poisson laws of parameters $m(n\pm 1)/2$. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line.